insight - Computational Complexity - # Chromatic Polynomial Zeros and Ground State Entropy of Potts Antiferromagnet on Hanoi Graphs

Core Concepts

The authors study the properties of the Potts model partition function and chromatic polynomials on the m-th iterates of Hanoi graphs, Hm, and use the results to draw inferences about the m→∞ limit that yields the self-similar Hanoi fractal, H∞. They calculate the ground state degeneracy per vertex of the zero-temperature Potts antiferromagnet on Hm and estimate the values for H∞. The authors also compute the zeros of the Potts partition function in the complex q-plane and complex v-plane, and analyze their accumulation in the m→∞ limit.

Abstract

The authors investigate the properties of the Potts model partition function Z(Hm, q, v) and chromatic polynomials P(Hm, q) on the m-th iterates of Hanoi graphs, Hm. They use an iterative procedure to calculate these quantities up to m=4.

From the calculations of the ground state degeneracy per vertex, W(Hm, q), for the Potts antiferromagnet on Hm, the authors extrapolate to estimate the values of W(H∞, 3) and W(H∞, 4). These are compared with known results for other lattices and hierarchical graphs.

The authors also study the zeros of the chromatic polynomials P(Hm, q) in the complex q-plane. They infer that in the m→∞ limit, the accumulation locus Bq crosses the real q-axis at a maximal point qc = (1/2)(3+√5), which is equal to the Tutte-Beraha number B5. This value of qc is the same as the critical point for the Potts antiferromagnet on the honeycomb lattice.

Additionally, the authors analyze the partition function zeros in the complex y-plane for large q, showing that they approximately accumulate along parts of the sides of an equilateral triangle.

To Another Language

from source content

arxiv.org

Stats

The authors provide the following key numerical results:
W(H∞, 3) = 1.489(1)
W(H∞, 4) = 2.499(1)
qc(H∞) = (1/2)(3+√5) = 2.618034

Quotes

None.

Key Insights Distilled From

by Shu-Chiuan C... at **arxiv.org** 10-01-2024

Deeper Inquiries

The chromatic zero patterns of the Hanoi graphs exhibit distinct characteristics when compared to those of the Sierpinski gasket graphs. In the case of the Hanoi graphs, the chromatic zeros form a roughly oval shape centered around ( q = 1 ), with a notable absence of complex-conjugate pairs of zeros extending into the interior of the region defined by the outer envelope of zeros. This contrasts with the Sierpinski gasket graphs, where multiple zero-free regions exist, and complex-conjugate pairs of zeros can be found within the inner regions, indicating a more intricate structure of zeros.
Regarding ground state degeneracies, the estimates for the Potts antiferromagnet on the Hanoi graphs show a monotonically decreasing function of ( m ) for fixed ( q ), leading to values of ( W(H_\infty, 3) = 1.489(1) ) and ( W(H_\infty, 4) = 2.499(1) ). In comparison, the Sierpinski gasket graphs have a ground state degeneracy of ( W(S_\infty, 3) = 1 ) and ( W(S_\infty, 4) = 2.026346 ). The inequality ( W(H_m, q) > W(S_m, q) ) for ( m \geq 1 ) reflects the fact that the effective vertex degree of the Hanoi fractal is lower than that of the Sierpinski gasket, leading to greater coloring freedom and higher ground state degeneracy in the Hanoi graphs.

The implication of the Hanoi fractal sharing the same critical point ( q_c = (1/2)(3 + \sqrt{5}) ) as the honeycomb lattice, despite differing effective vertex degrees, suggests a deeper connection between the structural properties of these two graph families. Both the Hanoi fractal and the honeycomb lattice exhibit self-similarity, which may contribute to their critical behavior being characterized by the same ( q_c ).
This shared critical point indicates that, at ( q_c ), both systems undergo a phase transition, which is a critical feature of the Potts model. The effective vertex degree of the honeycomb lattice is ( \Delta = 3 ), while the Hanoi fractal has an effective vertex degree of ( \Delta_{\text{eff}} = 3 ) as well. This similarity in effective vertex degree, despite the structural differences, implies that the underlying mechanisms governing the phase transitions in these systems may be more universal than previously thought. It also raises questions about the role of fractal geometry in critical phenomena and suggests that other fractal structures may exhibit similar critical behavior.

Yes, the methods employed to analyze the partition function zeros on Hanoi graphs can indeed be extended to study zeros on other types of fractal or hierarchical lattices. The iterative procedures for calculating the Tutte polynomial and the corresponding partition function zeros, as demonstrated in the study of Hanoi graphs, provide a robust framework that can be adapted to various graph families.
For instance, similar techniques can be applied to the Sierpinski gasket graphs, as previously shown, and can also be utilized for other fractal structures such as the Diamond Hierarchical Lattices or even more complex fractals. The key lies in the ability to express the partition function in terms of contributions from spanning subgraphs, which is a common feature across many hierarchical and fractal graphs.
Moreover, the insights gained from the chromatic zeros and their accumulation patterns in the complex plane can be valuable for understanding the critical behavior of other systems. By leveraging the established relationships between the partition function, chromatic polynomials, and graph-theoretic properties, researchers can explore the critical points and phase transitions in a broader class of fractal and hierarchical lattices, potentially uncovering new phenomena and universal behaviors across different graph families.

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